Solve each inequality. Graph the solution set and state the...

Solve each inequality. Graph the solution set and state the
solution set using interval notation.
See Examples 1 and 2 See the Strategy for Solving Quadratic Inequalities with a Sign Graph box on page 565
$$
2 v^{2}+7 v<4
$$

Key Concepts

Quadratic Inequalities

Quadratic inequalities involve an expression of the form ax² + bx + c compared to zero. The goal is to determine the variable values that make the expression positive or negative. Solving these inequalities typically requires finding the roots of the corresponding quadratic equation and testing the sign of the expression over different intervals divided by these roots.

Finding the Zeros of a Quadratic Expression

Determining the zeros (or roots) of the quadratic expression, by either factoring or using the quadratic formula, is crucial. The zeros represent the points where the quadratic expression equals zero and serve as boundaries that divide the number line into intervals. Analyzing these intervals reveals where the inequality holds true.

Sign Chart Method

The sign chart method involves selecting test points from each interval created by the zeros of the quadratic expression. By evaluating the sign of the quadratic expression at these test points, one can determine whether the expression is positive or negative in the interval. This systematic approach helps in identifying the solution set for the inequality.

Number Line Graph Representation

Graphing the solution set on a number line provides a visual representation of the intervals where the inequality is satisfied. Critical points, usually the zeros of the quadratic, are marked on the number line, and the intervals that meet the inequality’s conditions are highlighted. This visual aid reinforces the understanding of the solution set.

Interval Notation

Interval notation is a concise way to represent sets of numbers that satisfy an inequality. It uses brackets and parentheses to denote closed and open endpoints, respectively. This notation communicates the range of values for which the inequality holds, making it a standardized method of presenting the solution set.

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